By the most direct or shortest route. Since crows normally fly straight to their food supply, this simile came into use as the shortest distance between two points. It originated in the late eighteenth century or even earlier.

To acknowledge an embarrassing error and humiliatingly abase oneself. All these expressions date from the early nineteenth century, eating crow from America and eating humble pie and dirt from Britain. The origin of the first is not known, although it is generally acknowledged that the meat of a crow tastes terrible. A story cited by Charles Funk and published in the Atlanta Constitution in 1888 claims that toward the end of the War of 1812, during a temporary truce, an American went hunting and by accident crossed behind the British lines, where he shot a crow. He was caught by an unarmed British officer who, by complimenting him on his fine shooting, persuaded him to hand over his gun. The officer then pointed the gun and said that as punishment for trespassing the American must take a bite out of the crow. The American obeyed, but when the officer returned his gun, he took his revenge and made the Briton eat the rest of the bird. The source of humble pie is less far-fetched; it is a corruption of (or pun on) umble-pie, “umbles” being dialect for the heart, liver, and entrails of the deer, which were fed to the hunt’s beaters and other servants while the lord and his guests ate the choice venison. This explanation appeared in 1830 in Vocabulary of East Anglia by Robert Forby. The analogy to eating dirt is self-evident. It appeared in Frederick W. Farrar’s Julian Home (1859): “He made up for the dirt they had been eating by the splendour of his entertainment.”